arXiv subjects

Saurabh Basu

Publications and source records attributed to Saurabh Basu.

At least 19 recordsLinked to original sources

arXiv2026

Sector-resolved non-Bloch topology and nonlocal entanglement dynamics in a bond-dissipative Kitaev chain

A core characteristic of dissipative non-Hermitian topology is that the relaxation dynamics tracks the non-Bloch bulk-boundary correspondence, rendering an algebraic decay in the gapless regime and an exponential falloff in the gapped phase, so that local observables directly diagnose the topology. We show that this correspondence breaks down in a dissipative topological superconductor, where the local observables turn blind to the very topology they are expected to decipher. Via a bond-dissipative dimerized Kitaev chain in a third-quantized rapidity-matrix formulation, we find that at zero chemical potential the Majorana rapidity matrix decomposes into two independent non-Hermitian sectors, each with its own generalized Brillouin zone and non-Bloch winding number, thereby revealing a sector-resolved non-Bloch bulk-boundary correspondence. The local density is a cross-sector covariance and relaxes at the sum of the two sector rates, so it remains sector-blind even when one sector is gapless and topological. For balanced gain and loss, the finite-time zero events of the entanglement spectrum under purely periodic-boundary Lindblad evolution recover this hidden edge content sector by sector, serving as a dynamical invariant that returns the open-boundary edge rapidities without physically opening the chain.

quant-ph↗

arXiv2026

Large nonlinear Hall effect in strained moir\'e structures hosting pseudospin-3/2 fermions

We investigate the linear and nonlinear Hall response of a moir\'e \emph{watermill lattice}, in which stacking and twisting generate a four-band manifold near the Fermi level with suppressed group velocities at discrete magic angles. Including an inversion symmetry breaking onsite mass breaks the interlayer symmetry, opening a gap in this manifold and driving the system into a non-trivial bulk topological phase. We map the resulting phase diagram as a function of the strength of the mass and twist angle $\theta$, revealing several sectors with high Chern numbers. We then introduce strain to break the residual $C_3$ symmetry of the lattice which activates a finite Berry curvature dipole and correspondingly, a nonlinear Hall response. The dipole reverses sign sharply across topological phase boundaries, producing butterfly like features when plotted against the relevant system parameters. Its magnitude substantially exceeds that reported for symmetry-broken transition metal dichalcogenides, consistent with the elevated Wilson-loop winding and enhanced quantum geometry associated with the lattice's pseudospin-$3/2$ character. We conclude by incorporating thermal effects on the Berry curvature dipole, asserting that it is an important tool for discerning topology at low temperatures.

cond-mat.mes-hall↗

arXiv2026

Probing topological phase transitions via nonlinear Hall response in strained moir\'e dice lattice

Valley polarized twisted bilayer dice lattice hosts topologically nontrivial flat bands far from charge neutrality due to broken time reversal symmetry, whereas the ones in the vicinity of it remain topologically trivial. However, when both valleys are taken into consideration, the time reversal symmetry is preserved, which poses a serious hindrance to enumerate the valley specific topological phases that rely on the detection of the Berry curvature. In this work, we demonstrate that such a twisted structure with an applied uniaxial strain exhibits a nonlinear Hall effect far from charge neutrality. We ascertain that the nonlinear anomalous Hall signals can serve as a probe for topological phase transitions associated with a specific energy state that is constrained to reside at the lower edge of the middle subband and controlled via a staggered mass. Specifically, we show that the nonlinear anomalous Hall response undergoes a sign reversal across the topological phase boundaries. By tuning the carrier density, we compute the nonlinear Hall response obtained from the Berry curvature dipole, both in the chiral limit, and also when the chiral symmetry is broken. It is further seen that the nonlinear Hall effect is significantly enhanced in the broken chiral symmetry regime.

cond-mat.mes-hall↗

arXiv2026

Emergence of Non-Hermitian Magic Angles and Topological Phase Transitions in Twisted Bilayer $\alpha$-$T_3$ Lattices

We investigate the flat-band properties and topological phase transitions in a non-Hermitian twisted bilayer $\alpha-T_3$ lattice. Here, non-Hermiticity is introduced via Hatano-Nelson-type asymmetric hopping, while an aligned hexagonal boron nitride substrate provides a staggered sublattice mass to the system. We find that the introduction of non-reciprocal hopping splits the conventional single magic angle into three distinct non-Hermitian magic angles (NHMAs). Unlike the exceptional magic angles driven by spectral singularities, these NHMAs host perfectly isolated flat bands where the real and imaginary parts of the bandwidth simultaneously vanish. By mapping the complex eigenspectrum across the moir\'e Brillouin zone, we show that the scattered energy eigenvalues coalesce into well-defined, closed loop-like structures as the non-Hermitian parameter strength increases, indicating emergence of a nontrivial point-gap topology and hence the non-Hermitian skin effect. Furthermore, we characterize the topological phases by computing the direct band gap and the biorthogonal Chern number. While the system exhibits a transition to a higher topological phase at weak non-Hermiticity, we demonstrate that stronger non-Hermiticity drives the gap-closing boundaries to merge and their topological charges to mutually annihilate. This convergence results in a trivial gap closing and a complete suppression of the intermediate topological phase, confirming that non-Hermiticity fundamentally plays a crucial role with regard to destabilizing the robust topological features of this moir\'e system.

cond-mat.mes-hall↗

arXiv2026

Second-order Skin Effect in a Brick-Wall Lattice

Non-Hermitian skin effect, which is a unique feature of non-Hermitian systems, exhibits the formation of an extensive number of boundary modes under open boundary conditions. However, its manifestation in higher dimensions remains elusive. In our work, we demonstrate a hybrid skin-topological effect arising from the interplay between first-order band topology and non-reciprocal hopping in an engineered two-dimensional brick-wall geometry. The non-Hermitian brick-wall lattice under open boundary conditions in both directions exhibits several unconventional spectral features. Notably, the eigenvalues associated with the corner skin modes do not exhibit non-trivial windings in the complex energy plane; instead, they exhibit dynamically stable exceptional point-like features that do not originate from eigenvector coalescence. In contrast, the remaining modes accumulate at the opposite pair. Of all the corner skin modes, only the two that originate from the topological corner states of the Hermitian brick-wall lattice remain localized at individual corners, while the rest accumulate at the pair of opposite corners. This spatial distribution contrasts sharply with the second-order skin effect, where corner skin modes are more uniformly distributed. Finally, for the non-Hermitian Brick-wall lattice, we design and implement the corresponding topolectrical circuit (circuit for a square lattice is included for comparison) to directly visualize the hybrid skin-topological modes.

cond-mat.mes-hall↗

arXiv2026

Floquet generation of hybrid-order topology and $\mathbb{Z}_2$-like bipolar localization

Periodic driving offers a powerful tool to engineer topological phases and even induce phase transitions that are inaccessible in static settings. In this work, we demonstrate that a suitable driving protocol applied to the Benalcazar-Bernevig-Hughes (BBH) model, a canonical quadrupolar insulator with a pi-flux-induced projective PT, gives rise to a hybrid-order topological phase in which the dispersive first-order edge states and localized higher-order corner modes are shown to coexist at distinct quasienergies. Further, extending the scenario to non-reciprocal hopping induced non-Hermiticity, we uncover a Z2-like skin effect characterized by a sharp transition from a unipolar to a bipolar eigenstate-localized phase. Remarkably, this phenomenon, which is established as a prerogative for spinful systems, emerges here purely from the interplay between the embedded gauge structure and Floquet-renormalized symmetry constraints, without invoking physical spin degrees of freedom. Further, the broken bulk-boundary correspondence in this driven non-Hermitian setting, can be restored via computing the two-dimensional generalized Brillouin zone (GBZ), which we construct through a symmetry-reduced mapping onto an effective one-dimensional problem. The resulting non-Bloch invariants faithfully capture both the higher-order topology and the unipolar to bipolar transition. These findings reveal periodic driving as a versatile and controllable handle for sculpting the interplay of higher-order topology, symmetry transmutation, and non-Hermitian skin physics in a single unified platform.

cond-mat.mes-hall↗

arXiv2026

Topological Magnons and Giant Orbital Nernst Effect in a Zigzag Kitaev Antiferromagnet

The exploration of topological and transport properties of collinear antiferromagnets and the role of Kitaev interactions in realising topological states therein have rarely been systematically addressed in literature. In this context, we consider a zigzag-ordered antiferromagnet with both extended Kitaev and Dzyaloshinskii-Moriya interactions (DMI) in presence of an external magnetic field to focus on the topological phases demonstrated by the magnon band structure and validated by the transport properties. The hybridization between the up- and down-spin sectors carries evidences of opening bulk gaps in the magnon band structure, giving rise to nontrivial topological phases characterized by finite Chern numbers, chiral edge modes, and a nonzero thermal Hall conductivity. Furthermore, generally speaking, a finite magnon orbital moment can exist and contribute to the Nernst response even when the net spin moment vanishes owing to the fundamental independence of the spin and orbital magnetizations. This motivates us to investigate the magnon orbital moment, orbital Berry curvature, and the resulting orbital Nernst conductivity associated with the magnon bands. We find that a giant orbital Nernst conductivity emerges even in the absence of an external magnetic field. Moreover, the distinction between different topological phases is more lucidly manifested via the orbital Nernst conductivity, thereby highlighting an enhanced sensitivity of the orbital transport to the underlying band topology. For completeness, we briefly discuss the scenario corresponding to a N\'eel-ordered spin alignment, which leads to a vanishing Chern number and consequently suppressed thermal Hall and orbital Nernst conductivities compared to the zigzag-ordered case, even in the presence of DMI and Kitaev interactions.

cond-mat.mes-hall↗

arXiv2025

Controlling Dissipative Topology Through Floquet Driving: From Transient Diagnostics to Boundary States Isolation

Engineering dissipative dynamics in open quantum systems is under active focus, especially in topological settings where resilient edge modes are expected to exhibit decay rates distinct from the bulk. In this letter, we propose an efficient dynamical scheme to discern such long-lived excitations. Employing a Floquet-Lindblad framework, we explore how periodic driving reshapes the key features of a paradigmatic topological model, namely a Creutz ladder. Our results bear testimony to a drive-induced unipolar-bipolar transition in the Liouvillian skin effect, which gets dynamically manifested as a chiral-helical damping crossover. Such a transition effectively rescales the bulk localization length, giving rise to a polarization drift that we identify as a new invariant for efficient diagnosis of the nontrivial phases. As the transition becomes more gradual via tuning drive-rescaled parameters, we uncover signatures of a scale-free localization where skin and extended modes co-exist with distinct decay rates. The emergent hierarchy of the decay rates yields two disparate timescales: a chiral wavefront that rapidly empties the bulk followed by a long-lived regime dominated by robust edge modes. Overall, our results provide convincing evidence that periodic driving serves as a powerful handle to manipulate dissipative topological phases and dynamically isolate the boundary modes.

cond-mat.mes-hall↗

arXiv2025

Emergent topology of flat bands in a twisted bilayer $\alpha$-$T_3$ lattice

We investigate an interesting interplay of destructive interference due to lattice geometry and band folding due to enlargement of the Brillouin zone in generating and subsequently modifying the band topology in a twisted bilayer $\alpha$-$T_3$ system. The pronounced degeneracy of the emergent flat band in the dice limit of the $\alpha$-$T_3$ lattice is removed on alignment with h-BN layers, resulting in the formation of sub-bands with varying topological characteristics. Remarkably, while the sub-band near charge neutrality exhibits a trivial behavior, a topologically non-degenerate singular sub-band emerges away from charge neutrality. The topological band remains isolated from the rest of the bands for a substantial area of the $\alpha - \theta$ plane (where $\alpha$ and $\theta$ correspond to the hopping ratio and twist angle respectively) while exhibiting multiple phase transitions as a function of the aforementioned parameters via hybridization with its nearest bands. We study the evolution of the hybrid Wannier charge center and the Chern number to characterize the different emergent topological phases. Finally, the degree of flatness of the topological band is studied as a function of both $\alpha$ and $\theta$ to explicitly show the influence of quantum interference and band folding on the width of the topological band.

cond-mat.mes-hall↗

arXiv2025

Floquet Non-Bloch Formalism for a Non-Hermitian Ladder: From Theoretical Framework to Topolectrical Circuits

Periodically driven systems intertwined with non-Hermiticity opens a rich arena for topological phases that transcend conventional Hermitian limits. The physical significance of these phases hinges on obtaining the topological invariants that restore the bulk-boundary correspondence, a task well explored for static non-Hermitian (NH) systems, while it remains elusive for the driven scenario. Here, we address this problem by constructing a generalized Floquet non-Bloch framework that analytically captures the spectral and topological properties of time-periodic NH systems. Employing a high-frequency Magnus expansion, we analytically derive an effective Floquet Hamiltonian and formulate the generalized Brillouin zone for a periodically driven quasi-one-dimensional system, namely, the Creutz ladder with a staggered complex potential. Our study demonstrates that the skin effect remains robust (despite the absence of non-reciprocal hopping) across a broad range of driving parameters, and is notably amplified in the low-frequency regime due to emergent longer-range couplings. We further employ a symmetric time frame approach that generates chiral-partner Hamiltonians, whose invariants, when appropriately combined, account for the full edge-state structure. To substantiate the theoretical framework, we propose a topolectrical circuit (TEC) that serves as a viable experimental setting. Apart from capturing the skin modes, the proposed TEC design faithfully reproduces the presence of distinct Floquet edge states, as revealed through the voltage and impedance profiles, respectively. Thus, our work not only offers a theoretical framework for exploring NH-driven systems, but also provides an experimentally feasible TEC architecture for realizing these phenomena stated above in a laboratory.

cond-mat.mes-hall↗

arXiv2025

Topological characterization of magnon-polaron bands and thermal Hall conductivity in a frustrated kagome antiferromagnet

Spin-phonon coupling and its efficacy in inducing multiple topological phase transitions in a frustrated kagome antiferromagnet have been rare in literature. To this end, we study the ramifications of invoking optical phonons in such a system via two different coupling mechanisms, namely, a local and a non-local one, which are distinct in their microscopic origin. In case of the local spin-phonon coupling, a single phonon mode affects the magnetic interactions, whereas in the non-local case, two neighbouring phonon modes are involved in the energy renormalization, and it would be worthwhile to compare and contrast between the two. To tackle these phonons, we propose an analytic approach involving a canonical spin-Peierls transformation applied to magnons. The formalism renders a hybridization between the magnons and the phonon modes, yielding magnon-polaron quasiparticles. In both the coupling regimes, validations for the topological signatures are systematically derived from the bulk and edge spectral properties of the magnon-polaron bands that are characterized by their corresponding Chern numbers. Thereafter, we investigate transitions from one topological phase to another solely via tuning the spin-phonon coupling strength. Moreover, these transitions significantly impact the behavior of the thermal Hall conductivity that aids in discerning distinct topological phases. Additionally, the explicit dependencies on the temperature and the external magnetic field are explored in inducing topological phase transitions associated with the magnon-polaron bands. Thus, our work serves as an ideal platform to probe the interplay of frustrated magnetism and polaronic physics.

cond-mat.mes-hall↗

arXiv2025

Emergent topological phases and coexistence of gapless and spectral-localized Floquet quantum spin Hall states via electron-phonon interaction

In this work, a thorough exploration has been carried out to unravel the role of electron-phonon interaction (EPI) in a Bernevig-Hughes-Zhang (BHZ) quantum spin Hall (QSH) insulator subjected to a time-periodic step drive. It is observed that upon inclusion of the EPI, the system demonstrates emergent Floquet QSH (FQSH) phases and several topological phase transitions therein, mediated solely by the interaction strength. Quite intriguingly, the emergence of topological zero ($\pi$) modes in the bulk that remains otherwise gapless in the vicinity of the $\pi$ (zero) energy sector is observed, thus serving as a prime candidate of robust topology in gapless systems. With other invariants being found to be deficient in characterizing such coexistent phases, a spectral localizer (SL) is employed, which distinctly ascertains the nature of the (zero or $\pi$) edge modes. Following the SL prescription, a real-space Chern marker computed by us further provides support to such \textit{gapless} Floquet topological scenario. Our results can be realized in advanced optical setups that may underscore the importance of EPI-induced Floquet features.

cond-mat.mes-hall↗

arXiv2025

Floquet-engineered diode performance in a Majorana-quantum dot Josephson junction

We study nonreciprocal signatures of Josephson current (JC) in a quantum dot (QD)-based Josephson junction (JJ) that comprises of two periodically driven Kitaev chains (KCs) coupled with an intervening QD. The simultaneous breaking of the inversion symmetry ($\mathcal{IS}$) and the time-reversal symmetry ($\mathcal{TRS}$), indispensable for the Josephson diode effect (JDE), is achieved solely via the two Floquet drives that differ by a finite phase, which eventually results in a nonreciprocal current, and hence yields a finite JDE. It may be noted that the Floquet Majorana modes generated at both the far ends of the KCs (away from the QD) and adjacent to the QD junctions mediate the JC owing to a finite superconducting (SC) phase difference in the two KCs. We calculate the time-averaged JC and inspect the tunability of the current-phase relation (CPR) to ascertain the diode characteristics. The asymmetric Floquet drive also manifests an anomalous JC signature in our KC-QD-KC JJ. Furthermore, additional control over the QD energy level can be achieved via an external gate voltage that renders flexibility for the Josephson diode (JD) to act as an SC switching device. Tuning different system parameters, such as the chemical potential of the KCs, Floquet frequency, the relative phase mismatch of the drives, and the gate voltage, our model shows the highest possible rectification to be around $70\%$. Summarizing, our study provides an alternative scenario, replacing the traditional usage of an external magnetic field and spin-orbit coupling effects in a JD via asymmetrically driven Kitaev leads that entail Majorana-mediated transport.

cond-mat.mes-hall↗

arXiv2025

Controlled probing of localization effects in the non-Hermitian Aubry-Andr\'e model via topolectrical circuits

Anderson localization and the non-Hermitian skin effect are two distinct confinement phenomena of the eigenfunctions that are driven, respectively, by disorder and nonreciprocity. Understanding their interplay within a unified framework offers valuable insights into the localization properties of low-dimensional systems. To this end, we investigate a non-Hermitian version of the celebrated Aubry-Andr\'e model, which serves as an ideal platform due to its unique self-dual properties and ability to demonstrate a delocalization-localization transition in one dimension. Interestingly, in our setting, the competition between Anderson localization and the skin effect can be precisely controlled via the complex phase of the quasiperiodic disorder. Additionally, by analyzing the time evolution, we demonstrate that quantum jumps between the skin states and the Anderson-localized states occur in the theoretical model. Further, to gain support for our theoretical predictions in an experimental platform, we propose a topolectrical circuit featuring an interface that separates two distinct electrical circuit networks. The voltage profile of the circuit exhibits confinement at the interface, analogous to the skin effect, while the phenomenon of Anderson localization in the circuit can be perceived via a predicted localization behavior near the excitation node, rather than exhibiting sudden non-Hermitian jumps, as observed in the tight-binding framework. This interplay leads to a spatially tunable localization of the output voltage of the circuit. Our findings provide deeper insights into the controlled confinement of the eigenstates of the non-Hermitian Aubry-Andr\'e model by designing analogous features in topolectrical circuits, opening avenues in the fabrication of advanced electronic systems such as highly sensitive sensors and efficient devices for information transfer and communication.

cond-mat.dis-nn↗

arXiv2024

Topological characterization of a non-Hermitian ladder via Floquet non-Bloch theory

In this paper, we study a non-Hermitian (NH) ladder subjected to a variety of driving protocols. The driven system looses chiral symmetry (CS) whose presence is indispensable for its topological characterization. Further, the bulk boundary correspondence (BBC) gets adversely affected due to the presence of non-Hermitian skin effect (NHSE). Here, we present a formalism that not only retrieves the lost CS, but also restores the BBC via the construction of a generalized Brillouin zone (GBZ). Specifically, we employ delta and step drives to compare and contrast between them with regard to their impact on NHSE. Further, a widely studied harmonic drive is invoked in this context, not only for the sake of completeness, but its distinct computational framework offers valuable insights on the properties of out-of-equilibrium systems. While the delta and the harmonic drives exhibit unidirectional skin effect in the system, the step drive may show bi-directional skin effect. Also, there are specific points in the parameter space that are devoid of skin effect. These act as critical points that distinguish the skin modes to be localized at one boundary or the other. Moreover, for the computation of the non-Bloch invariants, we employ GBZ via a pair of symmetric time frames corresponding to the delta and the step drives, while a high-frequency expansion was carried out to deal with the harmonic drive. Finally, we present phase boundary diagrams that demarcate distinct NH phases obtained via tracking the trajectories of the exceptional points. These diagrams demonstrate a co-existence of the zero and $\pi$ energy modes in the strong NH limit and thus may be relevant for studies of Floquet time crystals.

cond-mat.mes-hall↗

arXiv2024

Conductance properties of an $\alpha$-$T_3$ Corbino disk

In this work, we investigate an $\alpha$-$T_3$ lattice in the form of a Corbino disk, characterized by inner and outer radii $R_1$ and $R_2$, threaded by a tunable magnetic flux. Through exact (analytic) solution of the stationary Dirac-Weyl equation, we compute the transmission probability of the carriers and hence obtain the conductance features for $0<\alpha \leq 1$ ($\alpha$ denotes the strength of the hopping between the central atom and one of the other two) which allows ascertaining the role of the flat band, alongwith scrutinizing the transport features from graphene to a dice lattice. Our results reveal periodic Aharonov-Bohm (AB) oscillations in the conductance, reminiscent of the utility of the Corbino disk as an electron pump. Further, these results are strongly influenced by parameters, such as, doping level, ratio of the inner and outer radii, magnetic flux, and $\alpha$. Additionally, complex quantum interference effect resulting in the possible emergence of higher harmonic modes and split-peak structures in the conductance, become prominent for smaller $\alpha$ values and larger ratios of the radii. We also find that, away from the charge-neutrality point (zero doping), the conductance oscillations are more pronounced and sensitive to the various parameters, with the corresponding behaviour largely governed via the evanescent wave transport. The Fano factor reveals distinct transport regimes, transitioning from Poissonian to pseudo-diffusive for $\alpha < 1$, and from ballistic to pseudo-diffusive for $\alpha = 1$. This setup, thus serves as a fertile ground for studying the generation of quantum Hall current and Aharonov-Bohm (AB) oscillations in a flat band system, alongwith demonstrating intricate appearance of higher harmonics in electron transport.

cond-mat.mes-hall↗

arXiv2024

Magnons on a dice lattice: topological features and transport properties

In this paper, we study the topological properties of magnons on a dice lattice, also known as the dual of a more widely studied kagome lattice. This structure has a central atom at the center of the honeycomb lattice, which leads to the formation of a flat band. Magnetic Hamiltonians associated with the magnon bands are scarcely studied in this flat band system, which motivates us on examining an interplay of different magnetic spin interactions, such as the Heisenberg exchange, Dzyaloshinskii-Moriya interaction (DMI), pseudodipolar interaction (PDI) and magnetocrystalline anisotropies in a dice lattice. In particular, the objective is to ascertain their roles in inducing various topological phases and the phase transitions therein. The competing effects of the DMI and the PDI in inducing transitions from either topological to topological or topological to trivial phases are noted and the corresponding results are supported via the magnon band structures, presence (or absence) of edge modes in a nanoribbon geometry, and the transport characteristic, namely the discontinuities in the thermal Hall conductivities. Meanwhile, the magnetocrystalline anisotropy also plays an intriguing role, where distinct (nonuniform) values at the different sublattice sites result in a richer topological landscape with Chern numbers $C = \pm 2$, $\pm 1$, and $0$, while a uniform anisotropy yields only $C = \pm 2$ and $0$. This discrepancy arises from the broken valley symmetry in the nonuniform case. Finally, we have enriched our understanding on the role of the flat band in impacting the topological features by comparing some of the key results with that of a honeycomb structure.

cond-mat.mes-hall↗

arXiv2024

Holstein polaron in a pseudospin-$1$ quantum spin Hall system: first and second order topological phase transitions

We theoretically propose the occurrence of a quantum spin Hall (QSH) and a second order topological phase transition (TPT) driven by electron-phonon (e-p) coupling in a pseudospin-$1$ fermionic system on an $\alpha$-$T_3$ lattice. Our model is formulated in the spirit of the Kane-Mele model modified by the Holstein Hamiltonian. The Lang-Firsov approach is employed to describe polarons reasonably well in the anti-adiabatic (high frequency) limit and to obtain an effective electronic Hamiltonian. It is shown that the system possesses topologically nontrivial phases up to a critical e-p coupling, $\lambda_c$ and are characterized by the helical QSH edge states along with a non-zero $\mathbb{Z}_2$ invariant for a certain range of $\alpha$. The topological phase vanishes beyond $\lambda_c$ and is accompanied by a bulk gap closing transition at $\lambda_c$, manifesting a TPT. We observe a more intriguing phenomenon for higher values of $\alpha$, where the system exhibits TPTs supported by two distinct gap closing transitions at $\lambda_{c_1}$ and $\lambda_{c_2}$, while a slim region at slightly lower values hosts a semi-metallic signature below $\lambda_{c_1}$. Subsequently, to explore more intricate features, we introduce a time reversal symmetry breaking magnetic field to trigger the formation of a second order topological phase. The magnetic field, by construction causes a boundary dependent gapping out of the edge states, consequently giving rise to robust corner modes in a tailored open boundary conditions. We justify the formation of the higher order phase by employing an appropriate invariant, namely the projected spin Chern number. Finally, we show that the e-p coupling significantly influences the corner modes (and also the real space energy bandstructure), corroborating a higher order TPT as we tune $\lambda$ beyond a critical value for a given value of $\alpha$.

cond-mat.mes-hall↗